Open Access
October 1999 Locally Contractive Iterated Function Systems
David Steinsaltz
Ann. Probab. 27(4): 1952-1979 (October 1999). DOI: 10.1214/aop/1022874823


An iterated function system on $\mathscr{X}\subset\mathbb{R}^d$ is defined by successively applying an i.i.d.sequence of random Lipschitz functions from to $\mathscr{X}$ to $\mathscr{X}$. This paper shows how $F _n = f_1\circ\ldots\circ f_n$ may converge even in the absence of the strong contraction conditions, for instance, Lipschitz constant smaller than 1 on average,which earlier work has required. Instead, it is posited that there be a region of contraction which compensates for the noncontractive or even expansive part of the functions. Applications to queues, to self-modifying random walks and to random logistic maps are given.


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David Steinsaltz. "Locally Contractive Iterated Function Systems." Ann. Probab. 27 (4) 1952 - 1979, October 1999.


Published: October 1999
First available in Project Euclid: 31 May 2002

zbMATH: 0974.37037
MathSciNet: MR1742896
Digital Object Identifier: 10.1214/aop/1022874823

Primary: 58F08
Secondary: 26A18 , 60J05

Keywords: iteration , logistic maps , Lyapunov functions , random attractors , single-server queues , Zeno’s walk

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 4 • October 1999
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